Title:Nonconvex penalization of switching control of partial differential equations

Abstract: This paper is concerned with optimal control problems for parabolic partial
differential equations with pointwise in time switching constraints on the
control. A standard approach to treat constraints in nonlinear optimization is
penalization, in particular using $L^1$-type norms. Applying this approach to
the switching constraint leads to a nonsmooth and nonconvex
infinite-dimensional minimization problem which is challenging both
analytically and numerically. Adding $H^1$ regularization or restricting to a
finite-dimensional control space allows showing existence of optimal controls.
First-order necessary optimality conditions are then derived using tools of
nonsmooth analysis. Their solution can be computed using a combination of
Moreau-Yosida regularization and a semismooth Newton method. Numerical examples
illustrate the properties of this approach.